Large band splitting with tunable spin polarization in the two-dimensional ferroelectric $\mathrm{Ga}XY$ family ($X=\mathrm{Se},\mathrm{Te}$; $Y=\mathrm{Cl},\mathrm{Br},\mathrm{I}$)

It has been generally accepted that the spin-orbit coupling effect in noncentrosymmetric materials leads to the band splitting and nontrivial spin polarization in the momentum space. However, in some cases, zero net spin polarization in the split bands may occurs, dubbed as the band splitting with vanishing spin polarization (BSVSP) effect, protected by non-pseudo-polar point group symmetry of the wave vector in the first Brillouin zone [K. Liu et al., Nat. Commun. 10, 5144 (2019)]. In this paper, by using first-principles calculations, we show that the BSVSP effect emerges in two-dimensional (2D) nonsymmorphic $\mathrm{Ga}XY$ ($X$= Se, Te; $Y$= Cl, Br, I) family, a new class of 2D materials having in-plane ferroelectricity. Taking the GaTeCl monolayer as a representative example, we observe the BSVSP effect in the split bands along the $X\text{−}M$ line located in the proximity of the conduction band minimum. By using the $\vec{k}·\vec{p}$ Hamiltonian derived based on the symmetry analysis, we clarify that such an effect originates from the cancellation of the local spin polarization, enforced by non-pseudo-polar $C_{2v}$ point group symmetry of the wave vector along the $X\text{−}M$ line. Importantly, we find that the spin polarization can be effectively induced by applying an external out-of-plane electric field, indicating that an electrically tunable spin polarization for spintronic applications is plausible.

Figure 1

(a, b) Atomic structure of the $\mathrm{Ga}XY$ ML corresponding to it symmetry operations viewed in the $x\text{−}y$ and $x\text{−}z$ planes, respectively. The unit cell of the crystal is indicated by the black dashed lines and characterized by $a$ and $b$ lattice parameters in the $x$ and $y$ directions. The red and black circles show the local sectors $A$ and $B$, respectively. The ${\mathrm{Ga}}_1, X_1$, and $Y_1$ indicate the Ga, $X$, and $Y$ atoms located on the sector $A$, while the ${\mathrm{Ga}}_2, X_2$, and $Y_2$ represent the Ga, $X$, and $Y$ atoms located on the sector $B$. (c) First Brillouin zone of the $\mathrm{Ga}XY$ ML crystal characterized by the high symmetry $\vec{k}$ points ($\mathrm{\Gamma }, Y, M$, and $X$). (d) Phonon spectrum of the GaTeCl ML as a representative of the $\mathrm{Ga}XY$ ML compounds calculated in the FBZ.

Figure 2

(a) Electronic band structures of the GaTeCl ML calculated with (blue lines) and without (black lines) including the SOC. Partial density of states projected to the atomic orbital for (b) Ga, (c) Te, and (d) Cl atoms. The black, red, blue, and green lines indicate the $s, p_x, p_y$, and $p_z$ orbitals, respectively.

Figure 3

Spin polarization projected to the spin-split bands calculated for $\vec{k}$ along $\mathrm{\Gamma }-X-M-Y-\mathrm{\Gamma }$ symmetry line. Color bars represent expectation values of spin component $〈S_x〉, 〈S_y〉$, and $〈S_z〉$.

Figure 4

Atomic decomposition of the spin polarization projected to the spin-split bands along the $X\text{−}M$ line. The upper, middle, and lower panels show the decomposition of the spin polarization on the Ga, Te, and Cl atomic pairs in the unit cell, respectively. Color bars represent expectation values of spin component $〈S_x〉, 〈S_y〉$, and $〈S_z〉$.

Figure 5

Evolution of the band structures under small compressive uniaxial strain along the $x$ (zigzag) direction: (a) equilibrium, (b) $-4\%$, and (d) $-8\%$. Introducing the strain significantly modifies the electronic band structures where the CBM shifts from the $k$ point at the $\mathrm{\Gamma }\text{−}Y$ line to that at the $X\text{−}M$ line as highlighted by the red lines. (d) Spin polarization projected to the spin-split bands along $\mathrm{\Gamma }\text{−}X\text{−}M\text{−}Y\text{−}\mathrm{\Gamma }$ symmetry line for the case of the compressive uniaxial strain of $-8\%$. Color bars represent expectation values of the spin components $〈S_x〉, 〈S_y〉$, and $〈S_z〉$.

Figure 6

Electric field effect on the spin polarization. (a) The spin polarization projected to the band structures of GaTeCl ML along the $X\text{−}M$ line an external out-of-plane electric electric, filed $E_z=1.0$ V/Å is shown. The colors indicate spin polarization components ($S_x, S_y, S_z$). (b) The calculated spin polarization as a function of out-of-plane electric field. The spin polarizations are calculated at certain the $\vec{k}$ point in the lowest conduction band along the $X\text{−}M$ line as indicated in black point in Fig. 6.